I ran into this problem the other day. The proof is supposed to be done by exhibiting an explicit bijection between two sets, without using induction, recurrence, or generating functions.

Denote by $\omega(n)$ the number of permutations $\sigma\in S_n$ so that $\sigma$ has a square root (that is, there exists $\tau\in S_n$ so that $\tau^2 = \sigma$). Prove that $\omega(2n+1) = (2n+1)\omega(2n)$.

@Gerry For some reason it stopped working for me earlier soon after I posted, but now works again. Anyway, you could try this new link.
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Ragib ZamanAug 23 '12 at 13:27

Say, what? The new link just gave me some incomprehensible photograph.
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Gerry MyersonAug 23 '12 at 23:17

@Gerry The link most likely expired, the website only gives temporary links for downloading the document. The original link is working fine from several computers I've tried, but just in case it still doesn't work I've sent the pdf to your work email.
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Ragib ZamanAug 24 '12 at 3:03